Grand antiprism | |
---|---|

(Schlegel diagram wireframe) | |

Type | Uniform 4-polytope |

Uniform index | 47 |

Cells | 100+200 (3.3.3) 20 ( 3.3.3.5) |

Faces | 20 {5} 700 {3} |

Edges | 500 |

Vertices | 100 |

Vertex figure | Sphenocorona |

Symmetry group | Ionic diminished Coxeter group [[10,2<sup>+</sup>,10]] of order 400 |

Properties | convex |

A net showing two disjoint rings of 10 antiprisms. 200 tetrahedra (yellow) are in face contact with the antiprisms and 100 tetrahedra (red) contact only other tetrahedra. |

In geometry, the **grand antiprism** or **pentagonal double antiprismoid** is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform 4-polytope, discovered in 1965 by Conway and Guy.^{ [1] }^{ [2] } Topologically, under its highest symmetry, the pentagonal antiprisms have *D _{5d}* symmetry and there are two types of tetrahedra, one with

- Pentagonal double antiprismoid Norman W. Johnson
- Gap (Jonathan Bowers: for grand antiprism)
^{ [3] }

20 stacked pentagonal antiprisms occur in two disjoint rings of 10 antiprisms each. The antiprisms in each ring are joined to each other via their pentagonal faces. The two rings are mutually perpendicular, in a structure similar to a duoprism.

The 300 tetrahedra join the two rings to each other, and are laid out in a 2-dimensional arrangement topologically equivalent to the 2-torus and the ridge of the duocylinder. These can be further divided into three sets. 100 face mate to one ring, 100 face mate to the other ring, and 100 are centered at the exact midpoint of the duocylinder and edge mate to both rings. This latter set forms a flat torus and can be "unrolled" into a flat 10×10 square array of tetrahedra that meet only at their edges and vertices. See figure below.

In addition the 300 tetrahedra can be partitioned into 10 disjoint Boerdijk–Coxeter helices of 30 cells each that close back on each other. The two pentagonal antiprism tubes, plus the 10 BC helices, form an irregular discrete Hopf fibration of the grand antiprism that Hopf maps to the faces of a pentagonal antiprism. The two tubes map to the two pentagonal faces and the 10 BC helices map to the 10 triangular faces.

The structure of the grand antiprism is analogous to that of the 3-dimensional antiprisms. However, the grand antiprism is the only convex uniform analogue of the antiprism in 4 dimensions (although the 16-cell may be regarded as a regular analogue of the *digonal antiprism *). The only nonconvex uniform 4-dimensional antiprism analogue uses pentagrammic crossed-antiprisms instead of pentagonal antiprisms, and is called the * pentagrammic double antiprismoid *.

The vertex figure of the grand antiprism is a sphenocorona or *dissected regular icosahedron*: a regular icosahedron with two adjacent vertices removed. In their place 8 triangles are replaced by a pair of trapezoids, edge lengths φ, 1, 1, 1 (where φ is the golden ratio), joined together along their edge of length φ, to give a tetradecahedron whose faces are the 2 trapezoids and the 12 remaining equilateral triangles.

12 ( 3.3.3) | 2 ( 3.3.3.5) | Dissected regular icosahedron |

The grand antiprism can be constructed by *diminishing* the 600-cell: subtracting 20 pyramids whose bases are three-dimensional pentagonal antiprisms. Conversely, the two rings of pentagonal antiprisms in the grand antiprism may be triangulated by 10 tetrahedra joined to the triangular faces of each antiprism, and a circle of 5 tetrahedra between every pair of antiprisms, joining the 10 tetrahedra of each, yielding 150 tetrahedra per ring. These combined with the 300 tetrahedra that join the two rings together yield the 600 tetrahedra of the 600-cell.

This diminishing may be realized by removing two rings of 10 vertices from the 600-cell, each lying in mutually orthogonal planes. Each ring of removed vertices creates a stack of pentagonal antiprisms on the convex hull. This relationship is analogous to how a pentagonal antiprism can be constructed from an icosahedron by removing two opposite vertices, thereby removing 5 triangles from the opposite 'poles' of the icosahedron, leaving the 10 equatorial triangles and two pentagons on the top and bottom.

(The snub 24-cell can also be constructed by another diminishing of the 600-cell, removing 24 icosahedral pyramids. Equivalently, this may be realized as taking the convex hull of the vertices remaining after 24 vertices, corresponding to those of an inscribed 24-cell, are removed from the 600-cell.)

Alternatively, it can also be constructed from the decagonal ditetragoltriate (the convex hull of two perpendicular nonuniform 10-10 duoprisms where the ratio of the two decagons are in the golden ratio) via an alternation process. The decagonal prisms alternate into pentagonal antiprisms, the rectangular trapezoprisms alternate into tetrahedra with two new regular tetrahedra (representing a non-corealmic triangular bipyramid) created at the deleted vertices. This is the only uniform solution for the p-gonal double antiprismoids alongside its conjugate, the pentagrammic double antiprismoid from the decagrammic ditetragoltriate.

600-cell | Grand antiprism |
---|---|

H_{4} Coxeter plane | |

20-gonal | |

H_{3} Coxeter plane (slight offset) | |

These are two perspective projections, projecting the polytope into a hypersphere, and applying a stereographic projection into 3-space.

Wireframe, viewed down one of the pentagonal antiprism columns. | with transparent triangular faces |

Orthographic projection Centered on hyperplane of an antiprism in one of the two rings. | 3D orthographic projection of 100 of 120 600-cell vertices and 500 edges {488 of 1/2 (3-Sqrt[5]) and 12 of 2/(3+Sqrt[5])}. |

- ↑ J.H. Conway and M.J.T. Guy:
*Four-Dimensional Archimedean Polytopes*, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965. (Michael Guy is son of Richard K. Guy) - ↑ Conway, 2008, p.402-403 The Grand Antiprism
- ↑ Klitzing, Richard. "4D convex polychora Grand antiprism".

In geometry, the **600-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the **C _{600}**,

In geometry, a **uniform 4-polytope** is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In geometry of 4 dimensions or higher, a **duoprism** is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an *n*-polytope and an *m*-polytope is an (*n*+*m*)-polytope, where *n* and *m* are 2 (polygon) or higher.

In geometry, the **rectified 600-cell** or **rectified hexacosichoron** is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices.

In geometry, the **great icosahedron** is one of four Kepler-Poinsot polyhedra, with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.

In geometry, the **snub 24-cell** or **snub disicositetrachoron** is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.

In four-dimensional geometry, a **runcinated 24-cell** is a convex uniform 4-polytope, being a runcination of the regular 24-cell.

In four-dimensional geometry, a **runcinated 120-cell** is a convex uniform 4-polytope, being a runcination of the regular 120-cell.

A **uniform polytope** of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In geometry, a **uniform 5-polytope** is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.

The **great duoantiprism** is the only uniform star-duoantiprism solution p=5, q=5/3, in 4-dimensional geometry. It has Schläfli symbol {5}⊗{5/3}, s{5}s{5/3} or ht_{0,1,2,3}{5,2,5/3}, Coxeter diagram , constructed from 10 pentagonal antiprisms, 10 pentagrammic crossed-antiprisms, and 50 tetrahedra.

In geometry of 4 dimensions, a **5-5 duoprism** or **pentagonal duoprism** is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two pentagons.

In geometry of 4 dimensions, a **8-8 duoprism** or **octagonal duoprism** is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two octagons.

In geometry of 4 dimensions, a **6-6 duoprism** or **hexagonal duoprism** is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two hexagons.

In geometry of 4 dimensions, a **10-10 duoprism** or **decagonal duoprism** is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two decagons.

In geometry of 4 dimensions, a **4-6 duoprism**, a duoprism and 4-polytope resulting from the Cartesian product of a square and a hexagon.

In geometry of 4 dimensions, a **4-8 duoprism**, a duoprism and 4-polytope resulting from the Cartesian product of a square and an octagon.

In geometry of 4 dimensions, a **6-8 duoprism**, a duoprism and 4-polytope resulting from the Cartesian product of a hexagon and an octagon.

*Kaleidoscopes: Selected Writings of H.S.M. Coxeter*, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6- (Paper 23) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes II*, [Math. Zeit. 188 (1985) 559-591] 2.8 The Grand Antiprism

- (Paper 23) H.S.M. Coxeter,
- Anomalous convex uniform polychoron - Model 47 , George Olshevsky.
- Klitzing, Richard. "4D uniform polytopes (polychora) gap".
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,
*The Symmetries of Things*2008, ISBN 978-1-56881-220-5 (Chapter 26) The Grand Antiprism - Grand Antiprism and Quaternions Mehmet Koca, Mudhahir Al-Ajmi, Nazife Ozdes Koca (2009); Mehmet Koca et al. 2009 J. Phys. A: Math. Theor. 42 495201

- In the Belly of the Grand Antiprism (middle section, describing the analogy with the icosahedron and the pentagonal antiprism)

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